Reliable, Efficient, and Robust A-Posteriori Estimates for Nonlinear Elliptic Equations: Applications in Porous Media Flow

  • Mitra, Koondanibha (Hasselt University)
  • Vohralik, Martin (INRIA Paris)

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In this work, we consider strongly monotone and Lipschitz-continuous nonlinear elliptic problems. We apply a finite element discretization in conjunction with an iterative linearization scheme, such as the fixed-point or the Newton iterations. In this setting, we derive a-posteriori error estimates that are robust with respect to the ratio of the continuity over monotonicity constants in the dual-energy norm invoked by the linearization iterations. This is linked to an orthogonal decomposition of the total error into a linearization error component and a discretization error component, which can be further used to adaptively stop the linearization iterations for efficient error balancing. The applications cover diverse physical phenomena such as flow through porous media, mean curvature flow, and biological processes. Numerical experiments for the time-discrete Richards equation illustrate the effectiveness of the theoretical estimates. The work is further generalized to include variational problems.